# Map

In mathematics and related technical fields, the term **map** or **mapping** is often a synonym for *function*. Along these lines, a *partial map* is a *partial function*, and a *total map* is a *total function*.

In many specific branches of mathematics, the term is used for a function with a specific property relevant to that branch, such as a *continuous function* in topology, a *linear transformation* in linear algebra, etc.

In formal logic, the term is sometimes used for a *functional predicate*, whereas a function is a model of such a predicate in set theory.

A mapping *m* which has domain *A* and codomain *B* can be denoted symbolically as

- .

If element *a* belongs to the domain *A* and if element *b* belongs to the codomain *B* and if mapping *m* maps element *a* to element *b*, this can be denoted symbolically as

- ,

"*m* maps *a* to *b*", which means the same as *m*(*a*) = *b*, "function *m* of *a* is *b*", in function notation. In such case, element *b* can be referred to as the image of element *a*.

For any element *a* in a mapping's domain, *a* has one and only one image. That is, there exists a *b* such that and if then *b* = *c*.

A mapping *m* is an **injective mapping** *m* : *A* ↪ *B* if each element of *A* is mapped to a different element of *B* (note the use of arrow with hook). Injective mappings have the property that the domain's cardinality is less than or equal to the codomain's cardinality: card *A* ≤ card *B*.

A mapping *m* is a **surjective mapping** *m* : *A* ↠ *B* if every element of *B* is an image of some element of *A* (note the use of two headed arrow): that is, if the mapping's range is equal to the mapping's codomain. Surjective mappings have the property that the domain's cardinality is greater than or equal to the codomain's cardinality: card *A* ≥ card *B*.

A **bijective mapping** is both injective and surjective, and has the property that the domain's cardinality equals the codomain's cardinality. *A mapping has an inverse mapping if and only if the mapping is bijective*. Why? If the mapping is bijective, then it is evident that it has an inverse. Now, suppose that a mapping has an inverse . This inverse can only be a mapping if every element of its domain *B* has an image, which can only be the case if mapping *m* is surjective. Also, the inverse can be a mapping only if every element of its domain has no more than one image, which can be the case only if mapping *m* is injective. So if mapping *m* has an inverse mapping, then *m* must be bijective.

A mapping's domain and codomain are uniquely defined, so that if *m* : *A* → *B* is some mapping, and if *n* : *B* ↪ *C* is an inclusion mapping, *C* being a proper superset of *B*, then the mapping *n* o *m* which is their composite is a different mapping from *m*.

*See also*:

cs:Zobrazení (matematika) he:מיפוי (מתמטיקה) ja:写像 nl:Afbeelding (wiskunde) sl:Preslikava zh:映射